Strict monotonicity of weighted inhomogeneous percolation thresholds under covering maps

We study Bernoulli bond percolation under weak covering maps for a class of weighted inhomogeneous product measures such that each edge $e$ carries a positive weight $w(e)$ and is open at intensity $\lambda \ge 0$ with probability $1-\exp (-\lambda w(e))$. We prove a strict monotonicity of the weighted critical intensities ${\lambda }_c$ in the covering compatibility setting under bounded degree, uniform ellipticity, tame fibre and disjoint tree-lifting. While for the weighted uniform connection thresholds ${\lambda }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}$, we establish their strict monotonicity for non-injective weak covering maps under bounded degree, covering compatibility, uniform ellipticity, bounded-fibre and disjoint tree-lifting. Moreover, we verify a general-profile theorem on strict monotonicities of the two thresholds ${\lambda }_c$ and ${\lambda }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}}$ respectively under explicit compact-interval finite-energy, derivative bounds and aforementioned geometric conditions.